what do you get if you right click the game title and select "view game cd key"? If you get a cdkey from there, you can try and use that key when you download the linux game.
Also another thing that comes to mind is that you can go to
https://shelf.frictionalgames.com/
and try your email and cdkey and see if that works out, from there you can download the linux version.
Also I can upload the linux game to a file sharing site if you want AND if Frictional doesn't mind me doing it (I have it right here, purchased it a couple days ago, you will have to wait a couple of hours though till upload finishes).
You can also install steam on linux via wine and try to run the game directly from steam, most likely it will work pretty good.
Hope it helped